### 63603 results for qubit oscillator frequency

Contributors: Saito, S., Meno, T., Ueda, M., Tanaka, H., Semba, K., Takayanagi, H.

Date: 2005-08-19

We performed a spectroscopy measurement of the **qubit** with long (50 ns) single-**frequency** microwave pulses. We observed multi-photon resonant peaks ( Φ q b 1.5 Φ 0 ) in the dependence of P s w on f M W 1 at a fixed magnetic flux Φ q b . We obtained the **qubit** energy diagram by plotting their positions as a function of Φ q b / Φ 0 (Fig. Fig2(a)). We took the data around the degeneracy point Φ q b ≈ 1.5 Φ 0 by applying an additional dc pulse to the microwave line to shift Φ q b away from 1.5 Φ 0 just before the readout, because the dc-SQUID could not distinguish the **qubit** states around the degeneracy point. The top solid curve in Fig. Fig2(a) represents a numerical fit to the resonant **frequencies** of one-photon absorption. From this fit, we obtain the **qubit** parameters E J / h = 213 GHz, Δ / 2 π = 1.73 GHz, and α = 0.8. The other curves in Fig. Fig2(a) are drawn by using these parameters for n 1 = 2, 3, and 4....Next, we used short single-**frequency** microwave pulses with a **frequency** of 10.25 GHz to observe the coherent quantum dynamics of the **qubit**. Figures Fig2(b) and (c) show one- and four-photon Rabi **oscillations** observed at the operating points indicated by arrows in Fig. Fig2(a) with various microwave amplitudes V M W 1 . These data can be fitted by damped **oscillations** ∝ exp - t p / T d cos Ω R a b i t p , except for the upper two curves in Fig. Fig2(b). Here, t p and T d are the microwave pulse length and **qubit** decay time, respectively. To obtain Ω R a b i , we performed a fast Fourier transform (FFT) on the curves that we could not fit by damped **oscillations**. Although we controlled the **qubit** environment, there were some unexpected resonators coupled to the **qubit**, which could be excited by the strong microwave driving or by the Rabi **oscillations** of the **qubit**. We consider that these resonators degraded the Rabi **oscillations** in the higher V M W 1 range of Fig. Fig2(b). Figure Fig2(d) shows the V M W 1 dependences of Ω R a b i / 2 π up to four-photon Rabi **oscillations**, which are well reproduced by Eq. ( eq2). Here, we used only one scaling parameter a (10.25 GHz) = 0.013 defined as a f M W 1 ≡ 4 g 1 α 1 / ω M W 1 V M W 1 , because it is hard to measure the real amplitude of the microwave applied to the **qubit** at the sample position. The scaling parameter a f M W 1 reflects the way in which the applied microwave is attenuated during its transmission to the **qubit** and the efficiency of the coupling between the **qubit** and the on-chip microwave line. In this way, we can estimate the real microwave amplitude and the interaction energy between the **qubit** and the microwave 2 ℏ g 1 α 1 by fitting the dependence of Ω R a b i / 2 π on V M W 1 . These results show that we can reach a driving regime that is so strong that the interaction energy 2 ℏ g 1 α 1 is larger than the **qubit** transition energy ℏ ω q b ....Experimental results with single-**frequency** microwave pulses. (a) Spectroscopic data of the **qubit**. Each set of the dots represents the resonant **frequencies** f r e s caused by the one to four-photon absorption processes. The solid curves are numerical fits. The dashed line shows a microwave **frequency** f M W 1 of 10.25 GHz. (b) One-photon Rabi **oscillations** of P s w with exponentially damped **oscillation** fits. Both the **qubit** Larmor **frequency** f q b and the microwave **frequency** f M W 1 are 10.25 GHz. The external flux is Φ q b / Φ 0 = 1.4944. (c) Four-photon Rabi **oscillations** when f q b = 41.0 GHz, f M W 1 = 10.25 GHz, and Φ q b / Φ 0 = 1.4769. (d) The microwave amplitude dependence of the Rabi **frequencies** Ω R a b i / 2 π up to four-photon Rabi **oscillations**. The solid curves represent theoretical fits. Fig2...Experimental results with two-**frequency** microwave pulses. (a) [(b)] Two-photon Rabi oscillations due to a parametric process when f q b = f M W 2 + - f M W 1 . The solid curves are fits by exponentially damped oscillations. (c) [(d)] Rabi **frequencies** as a function of V M W 1 , which are obtained from the data in Fig. Fig3(a) [(b)]. The dots represent experimental data when V M W 2 = 16.9, 23.5, 33.0, and 52.0 [50.1, 62.9, 79.1, and 124.7] mV from the bottom set of dots to the top one. The solid curves represent Eq. ( eq3). The inset is a schematic of the parametric process that causes two-photon Rabi oscillation when f q b = f M W 2 + - f M W 1 . Fig3...Next, **we **used short single-frequency microwave pulses with a frequency of 10.25 GHz** to** observe the coherent quantum dynamics of the qubit. Figures Fig2(b) and (c) show one- and four-photon Rabi oscillations observed at the operating points indicated by arrows in Fig. Fig2(a) with various microwave amplitudes V M W 1** . **These data can be fitted by damped oscillations ∝ exp - t p / T d cos Ω R a b** i** t p , except for the upper two curves in Fig. Fig2(b). Here, t p and T d are the microwave pulse length and qubit decay time, respectively. To obtain Ω R a b** i** , **we **performed a fast Fourier transform (FFT) on the curves that **we **could not fit by damped oscillations. Although **we **controlled the qubit environment, there were some unexpected resonators coupled** to** the qubit, which could be excited by the strong microwave driving or by the Rabi oscillations of the qubit. We consider that these resonators degraded the Rabi oscillations in the higher V M W 1 range of Fig. Fig2(b). Figure Fig2(d) shows the V M W 1 dependences of Ω R a b** i** / 2 π up** to** four-photon Rabi oscillations, which are well reproduced by Eq. ( eq2). Here, **we **used only one scaling parameter a (10.25 GHz) = 0.013 defined as a f M W 1 ≡ 4 g 1 α 1 / ω M W 1 V M W 1 , because** it** is hard** to** measure the real amplitude of the microwave applied** to** the qubit at the sample position. The scaling parameter a f M W 1 reflects the way in which the applied microwave is attenuated during its transmission** to** the qubit and the efficiency of the coupling between the qubit and the on-chip microwave line. In this way, **we **can estimate the real microwave amplitude and the interaction energy between the qubit and the microwave 2 ℏ g 1 α 1 by fitting the dependence of Ω R a b** i** / 2 π on V M W 1** . **These results show that **we **can reach a driving regime that is so strong that the interaction energy 2 ℏ g 1 α 1 is larger than the qubit transition energy ℏ ω q b ....The measurements were carried out in a dilution refrigerator. The sample was mounted in a gold plated copper box that was thermalized** to** the base temperature of 20 mK ( k B T two-frequency microwave pulses, **we **added two microwaves MW1 and MW2 with frequencies of f M W 1 and f M W 2 , respectively by** using** a splitter SP (Fig. Fig1(b)). Then **we **shaped them into microwave pulses through two mixers. We measured the amplitude of MW k V M W k at the point between the attenuator and the mixer with an oscilloscope. We confirmed that unwanted higher-order frequency components in the pulses, for example | f M W 1 ± f M W 2 | , 2 f M W 1 , and 2 f M W 2 are negligibly small under our experimental conditions. First, **we **choose the operating point by setting Φ q b around 1.5 Φ 0 , which fixes the qubit Larmor frequency f q b** . **The qubit is thermally initialized** to** be in | g by waiting for 300 μ s, which is much longer than the qubit energy relaxation time (for example 3.8 μ s at f q b = 11.1 GHz). Then a qubit operation is performed by applying a microwave pulse** to** the qubit. The pulse, with an appropriate length** t **p , amplitudes V M W k , and frequencies f M W k , prepares a qubit in the superposition state of | g and | e** . **After the operation, **we **immediately apply a dc readout pulse** to** the dc-SQUID. This dc pulse consists of a short (70 ns) initial pulse followed by a long (1.5 μ s) trailing plateau that has 0.6 times the amplitude of the initial part. For Φ q b if the qubit is detected as being in | e , the SQUID switches** to** a voltage state and an output voltage pulse should be observed; otherwise there should be no output voltage pulse. By repeating the measurement 8000 times, **we **obtain the SQUID switching probability P s w , which is directly related** to** P e** t **p for the dc readout pulse with a proper amplitude. For Φ q b > 1.5 Φ 0 , P s w is directly related** to** 1 - P e** t **p ....(a) Scanning electron micrograph of a flux **qubit** (inner loop) and a dc-SQUID (outer loop). The loop sizes of the **qubit** and SQUID are 10.2 × 10.4 μ m 2 and 12.6 × 13.5 μ m 2 , respectively. They are magnetically coupled by the mutual inductance M ≈ 13 pH. (b) A circuit diagram of the flux **qubit** measurement system. On-chip components are shown in the dashed box. L ≈ 140 pH, C ≈ 9.7 pF, R I 1 = 0.9 k Ω , R V 1 = 5 k Ω . Surface mount resistors R I 2 = 1 k Ω and R V 2 = 3 k Ω are set in the sample holder. We put adequate copper powder filters CP and LC filters F and attenuators A for each line. Fig1...We performed a spectroscopy measurement of the qubit with long (50 ns) single-frequency microwave pulses. We observed multi-photon resonant peaks ( Φ q b 1.5 Φ 0 ) in the dependence of P s w on f M W 1 at a fixed magnetic flux Φ q b** . **We obtained the qubit energy diagram by plotting their positions as a function of Φ q b / Φ 0 (Fig. Fig2(a)). We took the data around the degeneracy point Φ q b ≈ 1.5 Φ 0 by applying an additional dc pulse** to** the microwave line** to** shift Φ q b away from 1.5 Φ 0 just before the readout, because the dc-SQUID could not distinguish the qubit states around the degeneracy point. The top solid curve in Fig. Fig2(a) represents a numerical fit** to** the resonant frequencies of one-photon absorption. From this fit, **we **obtain the qubit parameters E J / **h** = 213 GHz, Δ / 2 π = 1.73 GHz, and α = 0.8. The other curves in Fig. Fig2(a) are drawn by** using** these parameters for n 1 = 2, 3, and 4....Parametric control of a superconducting flux **qubit**...Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi oscillations stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi oscillations corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena....The measurements were carried out in a dilution refrigerator. The sample was mounted in a gold plated copper box that was thermalized to the base temperature of 20 mK ( k B T **frequency** microwave pulses, we added two microwaves MW1 and MW2 with **frequencies** of f M W 1 and f M W 2 , respectively by using a splitter SP (Fig. Fig1(b)). Then we shaped them into microwave pulses through two mixers. We measured the amplitude of MW k V M W k at the point between the attenuator and the mixer with an oscilloscope. We confirmed that unwanted higher-order **frequency** components in the pulses, for example | f M W 1 ± f M W 2 | , 2 f M W 1 , and 2 f M W 2 are negligibly small under our experimental conditions. First, we choose the operating point by setting Φ q b around 1.5 Φ 0 , which fixes the **qubit** Larmor **frequency** f q b . The **qubit** is thermally initialized to be in | g by waiting for 300 μ s, which is much longer than the **qubit** energy relaxation time (for example 3.8 μ s at f q b = 11.1 GHz). Then a **qubit** operation is performed by applying a microwave pulse to the **qubit**. The pulse, with an appropriate length t p , amplitudes V M W k , and **frequencies** f M W k , prepares a **qubit** in the superposition state of | g and | e . After the operation, we immediately apply a dc readout pulse to the dc-SQUID. This dc pulse consists of a short (70 ns) initial pulse followed by a long (1.5 μ s) trailing plateau that has 0.6 times the amplitude of the initial part. For Φ q b **qubit** is detected as being in | e , the SQUID switches to a voltage state and an output voltage pulse should be observed; otherwise there should be no output voltage pulse. By repeating the measurement 8000 times, we obtain the SQUID switching probability P s w , which is directly related to P e t p for the dc readout pulse with a proper amplitude. For Φ q b > 1.5 Φ 0 , P s w is directly related to 1 - P e t p ....Experimental results with single-**frequency** microwave pulses. (a) Spectroscopic data of the **qubit**. Each set of the dots represents the resonant **frequencies** f r e s caused by the one to four-photon absorption processes. The solid curves are numerical fits. The dashed line shows a microwave **frequency** f M W 1 of 10.25 GHz. (b) One-photon Rabi oscillations of P s w with exponentially damped oscillation fits. Both the **qubit** Larmor **frequency** f q b and the microwave **frequency** f M W 1 are 10.25 GHz. The external flux is Φ q b / Φ 0 = 1.4944. (c) Four-photon Rabi oscillations when f q b = 41.0 GHz, f M W 1 = 10.25 GHz, and Φ q b / Φ 0 = 1.4769. (d) The microwave amplitude dependence of the Rabi **frequencies** Ω R a b i / 2 π up to four-photon Rabi oscillations. The solid curves represent theoretical fits. Fig2...We next investigated the coherent **oscillations** of the **qubit** through the parametric processes by using short two-**frequency** microwave pulses. Figure Fig3(a) [(b)] shows the Rabi **oscillations** of P s w when the **qubit** Larmor **frequency** f q b = 26.45 [7.4] GHz corresponds to the sum of the two microwave **frequencies** f M W 1 = 16.2 GHz, f M W 2 = 10.25 GHz [the difference between f M W 1 = 11.1 GHz and f M W 2 = 18.5 GHz] and the microwave amplitude of MW2 V M W 2 was fixed at 33.0 [50.1] mV. They are well fitted by exponentially damped **oscillations** ∝ exp - t p / T d cos Ω R a b i t p . The Rabi **frequencies** obtained from the data in Fig. 3(a) [(b)] are well reproduced by Eq. ( eq3) without any fitting parameters (Fig. Fig3(c) [(d)]). Here, we used Δ , which was obtained from the spectroscopy measurement (Fig. Fig2(a)) and used a (10.25 GHz) = 0.013 and a (16.2 GHz) = 0.0074 [ a (11.1 GHz) = 0.013 and a (18.5 GHz) = 0.0082], which had been obtained from Rabi **oscillations** by using single-**frequency** microwave pulses with each **frequency**. Those results provide strong evidence that we can achieve parametric control of the **qubit** with two-**frequency** microwave pulses....Experimental results with two-**frequency** microwave pulses. (a) [(b)] Two-photon Rabi **oscillations** due to a parametric process when f q b = f M W 2 + - f M W 1 . The solid curves are fits by exponentially damped **oscillations**. (c) [(d)] Rabi **frequencies** as a function of V M W 1 , which are obtained from the data in Fig. Fig3(a) [(b)]. The dots represent experimental data when V M W 2 = 16.9, 23.5, 33.0, and 52.0 [50.1, 62.9, 79.1, and 124.7] mV from the bottom set of dots to the top one. The solid curves represent Eq. ( eq3). The inset is a schematic of the parametric process that causes two-photon Rabi **oscillation** when f q b = f M W 2 + - f M W 1 . Fig3...We next investigated the coherent oscillations of the qubit through the parametric processes by** using** short two-frequency microwave pulses. Figure Fig3(a) [(b)] shows the Rabi oscillations of P s w when the qubit Larmor frequency f q b = 26.45 [7.4] GHz corresponds** to** the sum of the two microwave frequencies f M W 1 = 16.2 GHz, f M W 2 = 10.25 GHz [the difference between f M W 1 = 11.1 GHz and f M W 2 = 18.5 GHz] and the microwave amplitude of MW2 V M W 2 was fixed at 33.0 [50.1] mV. They are well fitted by exponentially damped oscillations ∝ exp - t p / T d cos Ω R a b** i** t p** . **The Rabi frequencies obtained from the data in Fig. 3(a) [(b)] are well reproduced by Eq. ( eq3) without any fitting parameters (Fig. Fig3(c) [(d)]). Here, **we **used Δ , which was obtained from the spectroscopy measurement (Fig. Fig2(a)) and used a (10.25 GHz) = 0.013 and a (16.2 GHz) = 0.0074 [ a (11.1 GHz) = 0.013 and a (18.5 GHz) = 0.0082], which had been obtained from Rabi oscillations by** using** single-frequency microwave pulses with each frequency. Those results provide strong evidence that **we **can achieve parametric control of the qubit with two-frequency microwave pulses....Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi **oscillations** stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi **oscillations** corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena. ... Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi **oscillations** stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi **oscillations** corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena.

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Contributors: Reuther, Georg M., Zueco, David, Hänggi, Peter, Kohler, Sigmund

Date: 2011-05-05

Time-resolved **qubit** readout via nonlinear Josephson inductance...Sketch of the flux **qubit** (blue) coupled to a dc-SQUID. The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 . The SQUID with Josephson inductance L J is shunted by a capacitance C . The **frequency** shift of the resulting harmonic **oscillator** (green) can be probed by external resonant ac-excitation A cos Ω a c t via the transmission line (black), in which the quantum fluctuations ξ i n q m t are also present....We propose a generalisation of dispersive **qubit** readout which provides the time evolution of a flux **qubit** observable. Our proposal relies on the non-linear coupling of the **qubit** to a harmonic **oscillator** with high **frequency**, representing a dc-SQUID. Information about the **qubit** dynamics is obtained by recording the **oscillator** response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent **qubit** **oscillations**. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity....eq:in-out-total1. In an experiment, this can be achieved by lock-in techniques which we mimic in the following way : First, we focus on the associated spectrum ξ o u t ω depicted in figure fig:**qubit**-osc-phase-spectrum(b). It reflects the **qubit** dynamics in terms of two sidebands around the central peak related to the **oscillator** **frequency**, here chosen as Ω = 10 ω q b . The dissipative influence of the environment, modelled by a transmission line (see figure fig:setup), is reflected in a broadening of this peak. The corresponding **oscillator** bandwidth is given as 2 α Ω , where α denotes the dimensionless damping strength; see app:QME. Here, we recall that the **oscillator** is driven resonantly by the external driving signal A cos Ω a c t , that is, Ω = Ω a c . In the time domain, the sidebands correspond to the phase-shifted signal ξ o u t t = A cos Ω t - ϕ e x p t with slowly time-dependent phase ϕ e x p t . In order to obtain this phase ϕ e x p t , we select the spectral data from a **frequency** window of size 2 Δ Ω centred at the **oscillator** **frequency** Ω , which means that ξ o u t ω is multiplied with a Gaussian window function exp - ω - Ω 2 / Δ Ω 2 . We choose for the window size the resonator bandwidth, Δ Ω = α Ω , which turns out to suppress disturbing contributions from the low-**frequency** **qubit** dynamics. Finally, we centre the clipped spectrum at zero **frequency** and perform an inverse Fourier transform to the time domain. If the phase shift φ e x t was constant, one could use a much smaller measurement bandwidth. Then the outcome of the measurement procedure would correspond to homodyne detection of a quadrature defined by the phase shift and yield a value ∝ cos φ e x p ....Sketch of the flux **qubit** (blue) coupled to a dc-SQUID. The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 . The SQUID with Josephson inductance L J is shunted by a capacitance C . The **frequency** shift of the resulting harmonic **oscillator** (green) can be probed by external resonant ac-excitation A cos Ω a c t via the transmission line (black), in which the quantum fluctuations ξ i n q **m t** are also present....We propose a generalisation of dispersive **qubit** readout which provides the time evolution of a flux **qubit** observable. Our proposal relies on the non-linear coupling of the **qubit** to a harmonic **oscillator** with high **frequency**, representing a dc-SQUID. Information about the **qubit** dynamics is obtained by recording the **oscillator** response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent **qubit** oscillations. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity....(a) Fidelity defect δ F = 1 - F for the phases ϕ t and ϕ e x p t and (b) time-averaged trace distance D ̄ between the density operators of a **qubit** with finite coupling to the **oscillator** and a reference **qubit** without **oscillator**. Both quantities are depicted for various coupling strengths g 2 in dependence of the **oscillator** **frequency** Ω . All other parameters are as in figure fig:**qubit**-osc-phase-spectrum....Time-resolved measurement of coherent **qubit** **oscillations** at the degeneracy point ϵ = 0 . The full **qubit**-**oscillator** state was simulated with the quantum master equation eq:blochredfield with N = 10 **oscillator** states and the parameters Ω = Ω a c = 10 ω q b , g 1 = 0.1 ω q b , g 2 = 0.01 ω q b , A = 1.0 ω q b . The dimensionless **oscillator** dissipation strength is α = 0.12 . The resonator bandwidth is given by 2 α Ω = 2.4 ω q b . (a) Lock-in amplified phase ϕ e x p t (dashed green lines), compared to the estimated phase ϕ t (solid red line) of the outgoing signal ξ o u t t . Here, ϕ t ∝ σ x t [cf. equation eq:dr-osc-phase], which is corroborated by the inset showing that σ x t performs **oscillations** with (angular) **frequency** ω q b . (b) Power spectrum ξ o u t ω for the resonantly driven **oscillator** (blue solid line). The sidebands stemming from the **qubit** dynamics are visible at **frequencies** Ω ± ω q b . In order to extract the phase information, we apply a Gaussian window function with respect to the **frequency** window of half-width Δ Ω = 1.2 ω q b , which turns out to be the optimal value for the measurement bandwidth....We consider a superconducting flux **qubit** coupled to a SQUID as sketched in figure fig:setup. The SQUID is modelled as a harmonic **oscillator**, which gives rise to the Hamiltonian...(a) Fidelity defect δ F = 1 - F for the phases ϕ t and ϕ **e **x p t and (b) time-averaged trace distance D ̄ between the density operators of a **qubit** with finite coupling to the **oscillator** and a reference **qubit** without **oscillator**. Both quantities are depicted for various coupling strengths g 2 in dependence of the **oscillator** **frequency** Ω . All other parameters are as in figure fig:**qubit**-osc-phase-spectrum....(a) Fidelity defect δ F = 1 - F for** the **phases ϕ t and ϕ e x p t and (b) time-averaged trace distance D ̄ between** the **density operators of a **qubit** with finite coupling to** the ****oscillator** and a reference **qubit** without **oscillator**. Both quantities are depicted for various coupling strengths g 2 in dependence of** the ****oscillator** **frequency** Ω . All other parameters are as in figure fig:**qubit**-osc-phase-spectrum....Time-resolved measurement of coherent **qubit** oscillations at the degeneracy point ϵ = 0 . The full **qubit**-**oscillator** state was simulated with the quantum master equation eq:blochredfield with N = 10 **oscillator** states and the parameters Ω = Ω a c = 10 ω q b , g 1 = 0.1 ω q b , g 2 = 0.01 ω q b , A = 1.0 ω q b . The dimensionless **oscillator** dissipation strength is α = 0.12 . The resonator bandwidth is given by 2 α Ω = 2.4 ω q b . (a) Lock-in amplified phase ϕ **e **x p t (dashed green lines), compared to the estimated phase ϕ t (solid red line) of the outgoing signal ξ o u t t . Here, ϕ t ∝ σ x t [cf. equation eq:dr-osc-phase], which is corroborated by the inset showing that σ x t performs oscillations with (angular) **frequency** ω q b . (b) Power spectrum ξ o u t ω for the resonantly driven **oscillator** (blue solid line). The sidebands stemming from the **qubit** dynamics are visible at frequencies Ω ± ω q b . In order to extract the phase information, we apply a Gaussian window function with respect to the **frequency** window of half-width Δ Ω = 1.2 ω q b , which turns out to be the optimal value for the measurement bandwidth....In figure fig:**qubit**-osc-fid(a) we depict the fidelity defect δ F = 1 - F between ϕ e x p t and ϕ t as a function of the **oscillator** **frequency** Ω = Ω a c for different quadratic coupling coefficients g 2 . As expected, the overall fidelity is rather insufficient for small **oscillator** **frequency** Ω **oscillator** bandwidth is too small to resolve the **qubit** dynamics, i.e., if ω q b **oscillator** **frequencies**, we again observe an increase of the fidelity defect, which occurs the sooner the smaller g 2 . This latter effect, which is only visible for the smallest value of g 2 in figure fig:**qubit**-osc-fid(a), is directly explained by a reduced maximum angular visibility of the phase ϕ t ∝ g 2 / Ω . Thus, figure fig:**qubit**-osc-fid(a) provides a pertinent indication for the validity frame of our central relation ...In figure fig:**qubit**-osc-fid(a) we depict** the **fidelity defect δ F = 1 - F between ϕ e x p t and ϕ t as a function of** the ****oscillator** **frequency** Ω = Ω a c for different quadratic coupling coefficients g 2 . As expected, the overall fidelity is rather insufficient for small **oscillator** **frequency** Ω , if** the ****oscillator** bandwidth is too small to resolve the **qubit** dynamics, i.e., if ω q b **ure **fig:**qubit**-osc-phase-spectrum, this still corroborates that Ω = 10 ω q b is a good choice. In** the **limit of large **oscillator** frequencies, we again observe an increase of** the **fidelity defect, which occurs** the **sooner** the **smaller g 2 . This latter effect, which is only visible for** the **smallest value of g 2 in figure fig:**qubit**-osc-fid(a), is directly explained by a reduced maximum angular visibility of** the **phase ϕ t ∝ g 2 / Ω . Thus, figure fig:**qubit**-osc-fid(a) provides a pertinent indication for** the **validity frame of our central relation ...If the **qubit** is only weakly coupled to the **oscillator**, and if the latter is driven only weakly, the ** qubit’s** time evolution is rather coherent (see section sec:sn on

**qubit**decoherence). For this scenario, figure fig:

**qubit**-osc-phase-spectrum(a) depicts the time-dependent phase ϕ t computed with the measurement relation...eq:in-out-total1. In an experiment, this can be achieved by lock-in techniques which we mimic in

**the**following way : First, we focus on

**the**associated spectrum ξ o u t ω depicted in figure fig:

**qubit**-osc-phase-spectrum(b). It reflects the

**qubit**dynamics in terms of two sidebands around

**the**central peak related to

**the**

**oscillator**

**frequency**, here chosen as Ω = 10 ω q b . The dissipative influence of

**the**environment, modelled by a transmission line (see figure

**fig:**setup), is reflected in a broadening of this peak. The corresponding

**oscillator**bandwidth is given as 2 α Ω , where α denotes

**the**dimensionless damping strength; see app:QME. Here, we recall that

**the**

**oscillator**is driven resonantly by

**the**external driving signal A cos Ω a c t , that is, Ω = Ω a c . In

**the**time domain, the sidebands correspond to

**the**phase-shifted signal ξ o u t t = A cos Ω t - ϕ e x p t with slowly time-dependent phase ϕ e x p t . In order to obtain this phase ϕ e x p t , we select

**the**spectral data from a

**frequency**window of size 2 Δ Ω centred at

**the**

**oscillator**

**frequency**Ω , which means that ξ o u t ω is multiplied with a Gaussian window function exp - ω - Ω 2 / Δ Ω 2 . We choose for

**the**window size

**the**resonator bandwidth, Δ Ω = α Ω , which turns out to suppress disturbing contributions from

**the**low-

**frequency**

**qubit**dynamics. Finally, we centre

**the**clipped spectrum at zero

**frequency**and perform an inverse Fourier transform to

**the**time domain. If

**the**phase shift φ e x t was constant, one could use a much smaller measurement bandwidth. Then

**the**outcome of

**the**measurement procedure would correspond to homodyne detection of a quadrature defined by

**the**phase shift and yield a value ∝ cos φ e x p . ... We propose a generalisation of dispersive

**qubit**readout which provides the time evolution of a flux

**qubit**observable. Our proposal relies on the non-linear coupling of the

**qubit**to a harmonic

**oscillator**with high

**frequency**, representing a dc-SQUID. Information about the

**qubit**dynamics is obtained by recording the

**oscillator**response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent

**qubit**

**oscillations**. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity.

Files:

Contributors: Xian-Ting Liang

Date: 2008-09-03

The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-**frequency** bath. The parameters are the same as in Fig. 1.
...The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the **frequency** ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz.
...The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-**frequency** bath in different values of Ω0, the other parameters are the same as in Fig. 1.
...The response functions of the Ohmic bath in (a) low and (c) medium frequencies and effective bath in (b) low and (d) medium frequencies. The parameters are the same as in Fig. 1. The cut-off frequencies for the two cases are taken according to Fig. 2.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO....The response functions of the Ohmic bath in (a) low and (c) medium **frequencies** and effective bath in (b) low and (d) medium **frequencies**. The parameters are the same as in Fig. 1. The cut-off **frequencies** for the two cases are taken according to Fig. 2.
...Decoherence and relaxation of **qubits** coupled to low- and medium-**frequency** Ohmic baths directly and via a harmonic **oscillator**...The sketch map on the low-, medium-, and high-**frequency** baths.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the oscillation **frequency** of the IHO. ... Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO.

Files:

Contributors: Gustavsson, Simon, Bylander, Jonas, Yan, Fei, Forn-Díaz, Pol, Bolkhovsky, Vlad, Braje, Danielle, Fitch, George, Harrabi, Khalil, Lennon, Donna, Miloshi, Jovi

Date: 2012-01-30

(a) Rabi **frequency** of **qubit** A, measured vs at = 0 . The driving field seen by the **qubit** contains two components: one is due to direct coupling to the antenna, the other is due to the coupling mediated by the resonator. (b) Rabi traces for a few of the data points in panel (a). The microwaves in the antenna have the same amplitude and **frequency** for all traces. (c) Direct coupling between the antenna and the **qubit**, extracted from measurements similar to the one shown in panel (a). The coupling depends only weakly on **frequency**. (d) Microwave current in the resonator, induced by a fixed microwave amplitude in the antenna. The black line is a fit to the square root of a Lorentzian, describing the **oscillation** amplitude of a harmonic **oscillator** with f r = 2.3 and Q = 100 ....(a) Rabi **frequency** of **qubit** A, measured vs at = 0 . The driving field seen by the **qubit** contains two components: one is due to direct coupling to** the **antenna, the other is due to** the **coupling mediated by** the **resonator. (b) Rabi traces for a few of** the **data points in panel (a). The microwaves in** the **antenna have** the **same amplitude and **frequency** for all traces. (c) Direct coupling between** the **antenna and the **qubit**, extracted from measurements similar to** the **one shown in panel (a). The coupling depends only weakly on **frequency**. (d) Microwave current in** the **resonator, induced by a fixed microwave amplitude in** the **antenna. The black line is a fit to** the **square root of a Lorentzian, describing** the **oscillation amplitude of a harmonic oscillator with f r = 2.3 and Q = 100 ....Driven dynamics and rotary echo of a **qubit** tunably coupled to a harmonic **oscillator**...(a) Circuit diagram of the **qubit** and the **oscillator**. The **qubit** state is encoded in currents circulating clockwise or counterclockwise in the **qubit** loop (blue arrow), while the mode of the harmonic **oscillator** is shown by the red arrows. (b) Spectrum for device A, showing the **qubit** and the harmonic **oscillator**. In addition, the two-photon **qubit** ( / 2 ) and the **qubit**+resonator ( + ) transitions are visible. (c) Flux induced in the **qubit** loop by the dc bias current . The black lines are parabolic fits. (d) First-order coupling between the **qubit** and the ground state of the harmonic **oscillator**, showing that the coupling is tunable by adjusting . The coupling is zero at = * , which is slightly offset from = 0 due to fabricated junction asymmetry. The derivative ε is calculated from the curves in panel (c). The **qubit** parameters are: I P = 175 for device A and I P = 180 for device B. The resonators have quality factors Q ≈ 100 . The right-hand axis is calculated using = 2.2 and C e f f = 2 C = 14 for both samples....Having determined the coupling coefficients, we turn to analyzing how the presence of the resonator influences the ** qubit’s** driven dynamics. Figure fig:Rabi(a) shows the extracted Rabi

**frequency**of

**qubit**A as a function of , measured at = 2.6 . We find that changes by a factor of five over the range of the measurement, which is surprising since both the amplitude and the

**frequency**of the microwave current in the antenna are kept constant. The data points were obtained by fitting Rabi

**oscillations**to decaying sinusoids, a few examples of Rabi traces for different values of are shown in...(a) Spectrum of device B. The spectral line at 2 is the resonator, whereas the

**qubit**tunnel coupling is Δ = 5.4 . (b) Rabi

**frequency**vs bias current , measured at f = 5.4 and = 0 and for two different microwave drive amplitudes . Similar to device A, the Rabi

**frequency**depends strongly on , and scales linearly with drive amplitude. The black lines are fits to Eqs. ( eq:drive, eq:Rabi), using the same coupling parameters for both sets of data. Note that the range of in fig:RabiLongT1(b) is several times larger than in fig:Rabi(a)....We have investigated two devices with similar layouts but slightly different parameters, both made of aluminum. Device A was designed and fabricated at MIT Lincoln Laboratory and device B was designed and fabricated at NEC. Figure fig:Sample(b) shows a spectroscopy measurement of device A versus applied flux, with the

**qubit**flux detuning defined as = Φ + Φ 0 / 2 and Φ 0 = h / 2 e . The

**qubit**

**frequency**follows = Δ 2 + ε 2 , where the tunnel coupling Δ = 2.6 is fixed by fabrication and the energy detuning ε = 2 I P / h is controlled by the applied flux Φ ( I P is the persistent current in the

**qubit**loop). The resonator

**frequency**is around 2.3 and depends only weakly on and . In addition, there are features visible at

**frequencies**corresponding to the sum of the

**qubit**and resonator

**frequencies**, illustrating the coherent coupling between the two systems ....(a) Decay envelopes of

**the**Rabi and rotary-echo sequences for device B, measured with = 65 . The solid lines are fits to eq:f. (b) Decay times for Rabi and rotary echo, extracted from fits similar to

**the**ones shown in panel (a). The dashed line shows

**the**upper limit set by

**qubit**energy relaxation. The dotted line marks

**the**position for

**the**decay envelope shown in panel (a). (c,d) Schematic diagrams describing

**the**two pulse sequences in (a) and (b). For rotary echo, the phase of

**the**microwaves is rotated by 180 ∘ during

**the**second half of

**the**sequence....We have investigated two devices with similar layouts but slightly different parameters, both made of aluminum. Device A was designed and fabricated at MIT Lincoln Laboratory and device B was designed and fabricated at NEC. Figure fig:Sample(b) shows a spectroscopy measurement of device A versus applied flux, with the

**qubit**flux detuning defined as = Φ + Φ 0 / 2 and Φ 0 = h / 2 e . The

**qubit**

**frequency**follows = Δ 2 + ε 2 , where the tunnel coupling Δ = 2.6 is fixed by fabrication and the energy detuning ε = 2 I P / h is controlled by the applied flux Φ ( I P is the persistent current in the

**qubit**loop). The resonator

**frequency**is around 2.3 and depends only weakly on and . In addition, there are features visible at frequencies corresponding to the sum of the

**qubit**and resonator frequencies, illustrating the coherent coupling between the two systems ....(a) Spectrum of device B. The spectral line at 2 is

**the**resonator, whereas the

**qubit**tunnel coupling is Δ = 5.4 . (b) Rabi

**frequency**vs bias current , measured at f = 5.4 and = 0 and for two different microwave drive amplitudes . Similar to device A, the Rabi

**frequency**depends strongly on , and scales linearly with drive amplitude. The black lines are fits to Eqs. ( eq:drive, eq:Rabi), using

**the**same coupling parameters for both sets of data. Note that

**the**range of in fig:RabiLongT1(b) is several times larger than in fig:Rabi(a)....Having determined the coupling coefficients, we turn to analyzing how the presence of the resonator influences the

**qubit**’s driven dynamics. Figure fig:Rabi(a) shows the extracted Rabi

**frequency**of

**qubit**A as a function of , measured at = 2.6 . We find that changes by a factor of five over the range of the measurement, which is surprising since both the amplitude and the

**frequency**of the microwave current in the antenna are kept constant. The data points were obtained by fitting Rabi oscillations to decaying sinusoids, a few examples of Rabi traces for different values of are shown in...Figures fig:Rabi(c) and fig:Rabi(d) show how the two drive components depend on microwave

**frequency**, measured by changing the static flux detuning to increase the

**qubit**

**frequency**[see...We have investigated the driven dynamics of a superconducting flux

**qubit**that is tunably coupled to a microwave resonator. We find that the

**qubit**experiences an oscillating field mediated by off-resonant driving of the resonator, leading to strong modifications of the

**qubit**Rabi

**frequency**. This opens an additional noise channel, and we find that low-

**frequency**noise in the coupling parameter causes a reduction of the coherence time during driven evolution. The noise can be mitigated with the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence....(a) Decay envelopes of the Rabi and rotary-echo sequences for device B, measured with = 65 . The solid lines are fits to eq:f. (b) Decay times for Rabi and rotary echo, extracted from fits similar to the ones shown in panel (a). The dashed line shows the upper limit set by

**qubit**energy relaxation. The dotted line marks the position for the decay envelope shown in panel (a). (c,d) Schematic diagrams describing the two pulse sequences in (a) and (b). For rotary echo, the phase of the microwaves is rotated by 180 ∘ during the second half of the sequence....We have investigated the driven dynamics of a superconducting flux

**qubit**that is tunably coupled to a microwave resonator. We find that the

**qubit**experiences an

**oscillating**field mediated by off-resonant driving of the resonator, leading to strong modifications of the

**qubit**Rabi

**frequency**. This opens an additional noise channel, and we find that low-

**frequency**noise in the coupling parameter causes a reduction of the coherence time during driven evolution. The noise can be mitigated with the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence....(a) Circuit diagram of the

**qubit**and

**the**oscillator. The

**qubit**state is encoded in currents circulating clockwise or counterclockwise in the

**qubit**loop (blue arrow), while

**the**mode of

**the**harmonic oscillator is shown by

**the**red arrows. (b) Spectrum for device A, showing the

**qubit**and

**the**harmonic oscillator. In addition, the two-photon

**qubit**( / 2 ) and the

**qubit**+resonator ( + ) transitions are visible. (c) Flux induced in the

**qubit**loop by

**the**dc bias current . The black lines are parabolic fits. (d) First-order coupling between the

**qubit**and

**the**ground state of

**the**harmonic oscillator, showing that

**the**coupling is tunable by adjusting . The coupling is zero at = * , which is slightly offset from = 0 due to fabricated junction asymmetry. The derivative ε is calculated from

**the**curves in panel (c). The

**qubit**parameters are: I P = 175 for device A and I P = 180 for device B. The resonators have quality factors Q ≈ 100 . The right-hand axis is calculated using = 2.2 and C e f f = 2 C = 14 for both samples....To further investigate how the presence of the resonator affects the

**qubit**dynamics at large detunings, we performed measurements on device B. Figure fig:RabiLongT1(a) shows a spectrum of that device, where the

**qubit**and the resonator mode ( f r = 2 ) are clearly visible. This device has a larger tunnel coupling ( Δ = 5.4 ), which allows us to operate the

**qubit**at large

**frequency**detuning from the resonator while still staying at ε d c = 0 , where the

**qubit**, to first order, is insensitive to flux noise . The

**qubit**-resonator detuning corresponds to several hundred linewidths of the resonator, which is the regime of most interest for quantum information processing ....Figure fig:RabiLongT1(b) shows the Rabi

**frequency**vs bias current of device B, measured at f = 5.4 and for two different values of the microwave drive current . Similarly to ... We have investigated the driven dynamics of a superconducting flux

**qubit**that is tunably coupled to a microwave resonator. We find that the

**qubit**experiences an

**oscillating**field mediated by off-resonant driving of the resonator, leading to strong modifications of the

**qubit**Rabi

**frequency**. This opens an additional noise channel, and we find that low-

**frequency**noise in the coupling parameter causes a reduction of the coherence time during driven evolution. The noise can be mitigated with the rotary-echo pulse sequence, which, for driven systems, is analogous to the Hahn-echo sequence.

Files:

Contributors: Strand, J. D., Ware, Matthew, Beaudoin, Félix, Ohki, T. A., Johnson, B. R., Blais, Alexandre, Plourde, B. L. T.

Date: 2013-01-03

(color online) (a-c) Experimental data showing sideband oscillations as a function** of** pulse duration vs. flux-drive **frequency**. The amplitude** of** the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations** of** sideband oscillations vs. drive **frequency**. Vertical white lines running through each plot indicate the **frequency** slices used in Fig. fig:FreqVsAmpl....Figure fig:FreqVsAmpl(a) shows linecuts of the experimental (black dots) and numerical (full red lines) chevrons. The linecuts are taken at the **frequency** ω F C corresponding to the maximum-visibility sideband **oscillations**, indicated by the full and dashed vertical lines in Fig. 3. The agreement between the experiments and simulations is excellent. In particular, the decay rate of the **oscillations** can be explained by the separately measured loss of the **qubit** and cavity and roughly corresponds to κ + γ 1 / 2 , where γ 1 is the bare transmon relaxation rate. This is expected for **oscillations** between states | e 0 and | g 1 . It also indicates that for these powers, the visibility loss can be completely attributed to damping. The lack of experimental points at pulse widths < 30 n s is a technical limit of the present configuration of our electronics that can be improved in future experiments.x x...First-order sideband transitions with flux-driven asymmetric transmon **qubits**...(color online) (a) Schematic of energy levels in a combined **qubit**-resonator system, showing first-order red sideband transition. (b) Optical microscope image with inset showing expanded view of one of the **qubits**. The terminations of the flux-bias lines for both **qubits** are visible, and they are used for both dc bias and FC signals. (c) Schematic of **qubit**-cavity layout and signal paths....(color online) Spectroscopy vs. flux for Q2 showing g-e (solid blue points) and e-f (hollow red points) transition frequencies. Blue and red lines correspond to numerical fits. Heavy black line shows bare cavity resonance **frequency**. Vertical dashed line indicates flux bias point for sideband measurements described in subsequent figures along with ac flux drive amplitude, 2 Δ Φ = 70.9 m Φ 0 , corresponding to 2 Δ ω g e / 2 π = 572 MHz, used in Figs. 3(c), 4(c)....We used a sample consisting of two asymmetric transmon **qubits** capacitively coupled to the voltage antinodes of a coplanar waveguide resonator [Fig. fig:schem(b, c)]. The cavity had a bare fundamental resonance **frequency** ω r / 2 π = 8.102 G H z and decay rate κ / 2 π = 0.37 M H z . **Qubit**-state measurements were performed in the high-power limit . The **qubits**, labeled Q1 and Q2, were designed to be identical, with mutual inductances to their bias lines of 1 p H for Q2 and 2 p H for Q1. The **qubits** were excited by microwave pulses sent through the resonator, and the flux lines were used for dc flux biasing of the **qubits** as well as the high-speed flux modulation pulses for exciting sideband transitions. The dc flux lines included cryogenic filters before connecting to a bias-T for joining to the ac flux line, which had 20 / 6 / 10 d B of attenuation at the 4 K / 0.7 K / 0.03 K plates. The distribution of cold attenuators and the flux-bias mutual inductances were chosen as a compromise to allow for a sufficient flux amplitude for high-speed modulation of the **qubit** energy levels with negligible Joule heating of the refrigerator while avoiding excessive dissipation coupled to the **qubits** from the flux-bias lines....Figure fig:FreqVsAmpl(a) shows linecuts** of** the experimental (black dots) and numerical (full red lines) chevrons. The linecuts are taken at the **frequency** ω F C corresponding to the maximum-visibility sideband oscillations, indicated by the full and dashed vertical lines in Fig. 3. The agreement between the experiments and simulations is excellent. In particular, the decay rate** of** the oscillations can be explained by the separately measured loss** of** the** qubit** and cavity and roughly corresponds to κ + γ 1 / 2 , where γ 1 is the bare transmon relaxation rate. This is expected for oscillations between states | e 0 and | g 1 . It also indicates that for these powers, the visibility loss can be completely attributed to damping. The lack** of** experimental points at pulse widths < 30 n s is a technical limit** of** the present configuration** of** our electronics that can be improved in future experiments.x x...(color online) Spectroscopy vs. flux for Q2 showing g-e (solid blue points) and e-f (hollow red points) transition **frequencies**. Blue and red lines correspond to numerical fits. Heavy black line shows bare cavity resonance **frequency**. Vertical dashed line indicates flux bias point for sideband measurements described in subsequent figures along with ac flux drive amplitude, 2 Δ Φ = 70.9 m Φ 0 , corresponding to 2 Δ ω g e / 2 π = 572 MHz, used in Figs. 3(c), 4(c)....We used a sample consisting** of** two asymmetric transmon qubits capacitively coupled to the voltage antinodes** of** a coplanar waveguide resonator [Fig. fig:schem(b, c)]. The cavity had a bare fundamental resonance **frequency** ω r / 2 π = 8.102 G H z and decay rate κ / 2 π = 0.37 M H z . Qubit-state measurements were performed in the high-power limit . The qubits, labeled Q1 and Q2, were designed to be identical, with mutual inductances to their bias lines** of** 1 p H for Q2 and 2 p H for Q1. The qubits were excited by microwave pulses sent through the resonator, and the flux lines were used for dc flux biasing** of** the qubits as well as the high-speed flux modulation pulses for exciting sideband transitions. The dc flux lines included cryogenic filters before connecting to a bias-T for joining to the ac flux line, which had 20 / 6 / 10 d B** of** attenuation at the 4 K / 0.7 K / 0.03 K plates. The distribution** of** cold attenuators and the flux-bias mutual inductances were chosen as a compromise to allow for a sufficient flux amplitude for high-speed modulation** of** the** qubit** energy levels with negligible Joule heating** of** the refrigerator while avoiding excessive dissipation coupled to the qubits from the flux-bias lines....We demonstrate rapid, first-order sideband transitions between a superconducting resonator and a **frequency**-modulated transmon **qubit**. The **qubit** contains a substantial asymmetry between its Josephson junctions leading to a linear portion of the energy band near the resonator **frequency**. The sideband transitions are driven with a magnetic flux signal of a few hundred MHz coupled to the **qubit**. This modulates the **qubit** splitting at a **frequency** near the detuning between the dressed **qubit** and resonator **frequencies**, leading to rates up to 85 MHz for exchanging quanta between the **qubit** and resonator....(color online) (a),(b),(c) Sideband oscillations corresponding to the white slices in Fig. fig:chevron(a-c). Experimental points correspond to black dots; numerical simulations (not fits) indicated by red lines. (d) Sideband oscillation **frequency** vs. flux drive amplitude (lower horizontal axis) or corresponding **frequency** modulation amplitude (upper horizontal axis). The dashed line shows a linear fit to the low **frequency** data points, while the red solid line indicates the theoretical dependence from the numerical simulations. The full black line shows the analytical sideband oscillation **frequency** from Eq. ( eq:H:t)....(color online) (a-c) Experimental data showing sideband oscillations as a function of pulse duration vs. flux-drive **frequency**. The amplitude of the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations of sideband oscillations vs. drive **frequency**. Vertical white lines running through each plot indicate the **frequency** slices used in Fig. fig:FreqVsAmpl....Figure fig:FreqVsAmpl(d) shows the sideband oscillation **frequency** Ω / 2 π extracted from the experimental linecuts (blackxdots) as a function** of** the flux-modulation amplitude Δ Φ . As expected from Eq. ( eq:H:t), whose prediction is given by the solid black line, the dependence** of** Ω with Δ Φ is linear at low amplitude and deviates at larger amplitudes. Beyond this simple model with only two transmon levels, quantitative agreement is found between the measured data and numerical simulations (full red line). For the numerical simulations, the link between the theoretical flux modulation amplitude Δ Φ and applied power is made by taking advantage** of** the linear dependence** of** Ω with Δ Φ at low power. Because** of** this, it is possible to convert the experimental flux amplitude from arbitrary units to m Φ 0 using only the lowest drive amplitude for calibration....Figure fig:FreqVsAmpl(d) shows the sideband **oscillation** **frequency** Ω / 2 π extracted from the experimental linecuts (blackxdots) as a function of the flux-modulation amplitude Δ Φ . As expected from Eq. ( eq:H:t), whose prediction is given by the solid black line, the dependence of Ω with Δ Φ is linear at low amplitude and deviates at larger amplitudes. Beyond this simple model with only two transmon levels, quantitative agreement is found between the measured data and numerical simulations (full red line). For the numerical simulations, the link between the theoretical flux modulation amplitude Δ Φ and applied power is made by taking advantage of the linear dependence of Ω with Δ Φ at low power. Because of this, it is possible to convert the experimental flux amplitude from arbitrary units to m Φ 0 using only the lowest drive amplitude for calibration....(color online) (a-c) Experimental data showing sideband **oscillations** as a function of pulse duration vs. flux-drive **frequency**. The amplitude of the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations of sideband **oscillations** vs. drive **frequency**. Vertical white lines running through each plot indicate the **frequency** slices used in Fig. fig:FreqVsAmpl....(color online) (a) Schematic of energy levels in a combined qubit-resonator system, showing first-order red sideband transition. (b) Optical microscope image with inset showing expanded view of one of the qubits. The terminations of the flux-bias lines for both qubits are visible, and they are used for both dc bias and FC signals. (c) Schematic of qubit-cavity layout and signal paths....(color online) (a),(b),(c) Sideband **oscillations** corresponding to the white slices in Fig. fig:chevron(a-c). Experimental points correspond to black dots; numerical simulations (not fits) indicated by red lines. (d) Sideband **oscillation** **frequency** vs. flux drive amplitude (lower horizontal axis) or corresponding **frequency** modulation amplitude (upper horizontal axis). The dashed line shows a linear fit to the low **frequency** data points, while the red solid line indicates the theoretical dependence from the numerical simulations. The full black line shows the analytical sideband **oscillation** **frequency** from Eq. ( eq:H:t). ... We demonstrate rapid, first-order sideband transitions between a superconducting resonator and a **frequency**-modulated transmon **qubit**. The **qubit** contains a substantial asymmetry between its Josephson junctions leading to a linear portion of the energy band near the resonator **frequency**. The sideband transitions are driven with a magnetic flux signal of a few hundred MHz coupled to the **qubit**. This modulates the **qubit** splitting at a **frequency** near the detuning between the dressed **qubit** and resonator **frequencies**, leading to rates up to 85 MHz for exchanging quanta between the **qubit** and resonator.

Files:

Contributors: Averin, Dmitri V., Rabenstein, Kristian, Semenov, Vasili K.

Date: 2005-10-27

(a) Equivalent circuit of the flux detector based on the Josephson transmission line (JTL) and (b) diagram of scattering of the fluxon injected into the JTL with momentum k by the potential U x that is controlled by the measured **qubit**. The fluxons are periodically injected into the JTL by the generator and their scattering characteristics (transmission and reflection coefficients t k , r k ) are registered by the receiver....Rapid Ballistic Readout for Flux **Qubits**...Schematics of the QND fluxon measurement of a **qubit** which suppresses the effect of back-action dephasing on the **qubit** **oscillations**. The fluxon injection **frequency** f is matched to the **qubit** **oscillation** **frequency** Δ : f ≃ Δ / π , so that the individual acts of measurement are done when the **qubit** density matrix is nearly diagonal in the σ z basis, and the measurement back-action does not introduce dephasing in the **oscillation** dynamics....Schematics of the QND fluxon measurement of a **qubit** which suppresses the effect of back-action dephasing on the **qubit** oscillations. The fluxon injection **frequency** f is matched to the **qubit** oscillation **frequency** Δ : f ≃ Δ / π , so that the individual acts of measurement are done when the **qubit** density matrix is nearly diagonal in the σ z basis, and the measurement back-action does not introduce dephasing in the oscillation dynamics. ... We suggest a new type of the magnetic flux detector which can be optimized with respect to the measurement back-action, e.g. for the situation of quantum measurements. The detector is based on manipulation of ballistic motion of individual fluxons in a Josephson transmission line (JTL), with the output information contained in either probabilities of fluxon transmission/reflection, or time delay associated with the fluxon propagation through the JTL. We calculate the detector characteristics of the JTL and derive equations for conditional evolution of the measured system both in the transmission/reflection and the time-delay regimes. Combination of the quantum-limited detection with control over individual fluxons should make the JTL detector suitable for implementation of non-trivial quantum measurement strategies, including conditional measurements and feedback control schemes.

Files:

Contributors: Jerger, Markus, Poletto, Stefano, Macha, Pascal, Hübner, Uwe, Il'ichev, Evgeni, Ustinov, Alexey V.

Date: 2012-05-29

In **the** next set **of** experiments, we tuned **the** uniform flux coil and two compact local coils placed above **the** sample to bias three qubits at their symmetry **points**. Limiting **the** number **of** qubits to three was necessary **because** **of** **the** lack **of** additional (on-chip) coils and not due to **the** readout technique itself. After setup **of** **the** readout pulse to probe circuits number #2, 3 and 5, we performed a spectroscopy **of** all three qubits simultaneously. A continuous microwave excitation signal **of** varying frequencies was applied to **the** sample and a pulsed three-tone probe signal was applied every 10 μ s . When **the** excitation **frequency** matches **the** gap between **the** ground and first excited states **of** a** qubit**,

**the**instantaneous dispersive shift

**of**

**the**center

**frequency**

**of**

**the**corresponding resonator switches between positive and negative, thus changing

**the**mean amplitude and phase

**of**

**the**transmitted probe tone. Figure fig:multi_spec shows

**the**spectra

**of**three qubits measured in parallel....Finally, we performed simultaneous manipulation with time resolved measurements on three qubits. Here, we used individual microwave excitations for every

**, which were added together via a power combiner. We note that**

**qubit****the**complete excitation chain could be replaced

**by**a reference source and a mixer controlled

**by**a single arbitrary waveform generator with sufficient bandwidth to drive all qubits, similar to FDM readout tone generation. Measurement data are reported in Fig. fig:rabi. All three qubits were simultaneously driven

**by**individual excitation tones and

**the**readout was performed in parallel using

**the**FDM protocol. Every

**can be Rabi-driven at a different power. Left panels**

**qubit****of**Fig. fig:rabi present Rabi oscillations at three different powers for all qubits. The measured linear power dependences

**of**Rabi oscillations reported on

**the**right panels in Fig. fig:rabi are in excellent agreement with theory....Finally, we performed simultaneous manipulation with time resolved measurements on three

**qubits**. Here, we used individual microwave excitations for every

**qubit**, which were added together via a power combiner. We note that the complete excitation chain could be replaced by a reference source and a mixer controlled by a single arbitrary waveform generator with sufficient bandwidth to drive all

**qubits**, similar to FDM readout tone generation. Measurement data are reported in Fig. fig:rabi. All three

**qubits**were simultaneously driven by individual excitation tones and the readout was performed in parallel using the FDM protocol. Every

**qubit**can be Rabi-driven at a different power. Left panels of Fig. fig:rabi present Rabi

**oscillations**at three different powers for all

**qubits**. The measured linear power dependences of Rabi

**oscillations**reported on the right panels in Fig. fig:rabi are in excellent agreement with theory....(color online). Simultaneous manipulation and detection of three

**qubits**. Left plots: Rabi oscillations at several powers; traces are vertically offset for better visibility; curves with the same color/offset (blue-bottom, green-center, red-top) are measured simultaneously using the FDM technique described in the main text. Right plots: Rabi oscillation

**frequency**versus power of the excitation tone; the error bars are smaller than the size of the dots....The transmission spectrum of the sample, measured with a vector network analyzer is reported in Fig. fig:tm6q(a). The seven absorption peaks correspond to the seven readout resonators. Close to each peak its bare resonance

**frequency**as well the identification number of the device are printed. The inset of Fig. fig:tm6q(b) shows the transmission at

**frequencies**around the resonance of device #3 vs. the magnetic flux bias. The two points at which the dispersive

**frequency**shift changes its sign correspond to avoided level crossings of the

**qubit**and resonator. We demonstrate FDM by measuring the maximum possible number of devices simultaneously. Figure fig:tm6q(b) shows the transmitted amplitude of the six probe tones versus the external uniformly applied magnetic flux. Each curve is shown aligned with the corresponding transmission peak to the left in Fig. fig:tm6q(a). The amplitude of the transmitted signal is constant as long as the

**qubit**remains far detuned from the resonator. The amplitude changes drastically around two distinct fluxes, again indicating anti-crossings between the

**qubit**and the corresponding resonator. There is a minimum between these two peaks, because the readout

**frequencies**were set on resonance, with the dispersive shifts at the symmetry points of the

**qubits**taken into account. The readout

**frequency**of device #3 is shown as a dashed line in the inset....

**Frequency**Division Multiplexing Readout and Simultaneous Manipulation of an Array of Flux

**Qubits**...superconducting flux

**qubit**,

**qubit**register, dispersive readout,

**frequency**division multiplexing, microwave resonators, Rabi oscillations ...(color online). (a) Transmission spectrum of the sample with all

**qubits**far detuned from the resonances. (b) FDM readout of six flux

**qubits**. The main plot shows the transmission amplitude at the resonance

**frequencies**of devices #1 to 6 vs. the magnetic flux generated by the uniform field coil, measured using FDM. The inset shows the transmission amplitude at several

**frequencies**close to resonance #3, measured with a network analyzer. The dashed line indicates the probe

**frequency**used for this device in the main plot. The curves in the main plot are normalized and shifted vertically for better visibility. The offset along the horizontal axis is due to magnetic field non-uniformity, which is likely due to the screening currents generated in the superconducting ground plane....Experimental setup used for FDM readout. The

**qubit**manipulation signal is generated by a single microwave source for spectroscopy and three additional microwave sources, DAC channels and mixers for pulsed excitation of the

**qubits**....The

**transmission**spectrum

**of**

**the**sample, measured with a vector network analyzer is reported in Fig. fig:tm6q(a). The seven absorption peaks correspond to

**the**seven readout resonators. Close to each peak its bare

**resonance**

**frequency**as well

**the**identification number

**of**

**the**device are printed. The inset

**of**Fig. fig:tm6q(b) shows

**the**

**transmission**at frequencies around

**the**

**resonance**

**of**device #3 vs.

**the**magnetic flux bias. The two

**points**at which

**the**dispersive

**frequency**shift changes its sign correspond to avoided level

**crossings**

**of**

**the**

**and resonator. We demonstrate FDM**

**qubit****by**measuring

**the**maximum possible number

**of**devices simultaneously. Figure fig:tm6q(b) shows

**the**transmitted amplitude

**of**

**the**six probe tones versus

**the**external uniformly applied magnetic flux. Each curve is shown aligned with

**the**corresponding

**transmission**peak to

**the**left in Fig. fig:tm6q(a). The amplitude

**of**

**the**transmitted signal is constant as

**long**as

**the**

**remains far detuned from**

**qubit****the**resonator. The amplitude changes drastically around two distinct fluxes, again indicating anti-

**crossings**between

**the**

**and**

**qubit****the**corresponding resonator. There is a minimum between these two peaks,

**because**

**the**readout frequencies were set on

**resonance**, with

**the**dispersive shifts at

**the**symmetry

**points**

**of**

**the**qubits taken into

**account**. The readout

**frequency**

**of**device #3 is shown as a dashed line in

**the**inset....An important desired ingredient of superconducting quantum circuits is a readout scheme whose complexity does not increase with the number of

**qubits**involved in the measurement. Here, we present a readout scheme employing a single microwave line, which enables simultaneous readout of multiple

**qubits**. Consequently, scaling up superconducting

**qubit**circuits is no longer limited by the readout apparatus. Parallel readout of 6 flux

**qubits**using a

**frequency**division multiplexing technique is demonstrated, as well as simultaneous manipulation and time resolved measurement of 3

**qubits**. We discuss how this technique can be scaled up to read out hundreds of

**qubits**on a chip....A probe signal composed of one microwave tone per

**qubit**to be read out is sent through the common transmission line. The interaction between each

**qubit**and resonator leads to a state-dependent dispersive shift, Δ ω r = g ~ 2 / ω q - ω r σ z , of the resonator

**frequency**, where g ~ is the effective coupling between the resonator and

**qubit**, ω q and ω r are the angular resonance

**frequencies**of the

**qubit**and resonator, and σ z is ± 1 depending on the state of the

**qubit**. The composite signal probes all resonators at the same time, storing the information on the state of all

**qubits**in the transmitted tones. Detection of the transmitted amplitude and phase of each of the tones provides a simultaneous non-destructive measurement of the states of all

**qubits**. The probe signal is generated by mixing a reference microwave tone in the band of the resonators and a multi-tone DAC output using an IQ mixer, see Fig. fig:setup. By using the I and Q quadratures, we address the upper and lower sidebands of the mixing product individually to effectively double the bandwidth of the system. The mixer output is combined with the

**qubit**manipulation signal through a directional coupler. A strongly attenuated line transmits the combined signal to the sample, which is attached to the mixing chamber stage of a dilution refrigerator. Two cryogenic circulators and a high-pass filter at 30 mK are used to prevent reflections and noise from traveling from the cryogenic amplifier back to the sample. A chain of amplifiers provides 80 dB gain to boost the transmitted probe signal to levels sufficient for the detection stage, which employs an identical IQ mixer to convert the signal back to baseband

**frequencies**. The local

**oscillator**inputs of both mixers are fed from the same reference microwave source, resulting in a homodyne detection with a fixed phase offset. An additional high-pass filter between the local

**oscillator**ports of the two mixers prevents leakage of the baseband signal. After digitizing both quadratures, the amplitude and phase of all components of the probe signal are extracted via FFT. The maximum number of devices that can be probed with the described technique is defined by the

**frequency**separation between resonators and the bandwidth of the acquisition board. The

**frequencies**of the resonators on our chip are spaced at intervals of 150 MHz and the acquisition board has a bandwidth somewhat below 500 MHz, allowing for the simultaneous detection of up to six devices....In the next set of experiments, we tuned the uniform flux coil and two compact local coils placed above the sample to bias three

**qubits**at their symmetry points. Limiting the number of

**qubits**to three was necessary because of the lack of additional (on-chip) coils and not due to the readout technique itself. After setup of the readout pulse to probe circuits number #2, 3 and 5, we performed a spectroscopy of all three

**qubits**simultaneously. A continuous microwave excitation signal of varying

**frequencies**was applied to the sample and a pulsed three-tone probe signal was applied every 10 μ s . When the excitation

**frequency**matches the gap between the ground and first excited states of a

**qubit**, the instantaneous dispersive shift of the center

**frequency**of the corresponding resonator switches between positive and negative, thus changing the mean amplitude and phase of the transmitted probe tone. Figure fig:multi_spec shows the spectra of three

**qubits**measured in parallel....(color online). Simultaneous manipulation and detection of three

**qubits**. Left plots: Rabi

**oscillations**at several powers; traces are vertically offset for better visibility; curves with the same color/offset (blue-bottom, green-center, red-top) are measured simultaneously using the FDM technique described in the main text. Right plots: Rabi

**oscillation**

**frequency**versus power of the excitation tone; the error bars are smaller than the size of the dots....A probe signal composed

**of**one microwave tone per

**to be read out is sent through**

**qubit****the**common

**transmission**line. The interaction between each

**and resonator leads to a state-dependent dispersive shift, Δ ω r = g ~ 2 / ω q - ω r σ z ,**

**qubit****of**

**the**resonator

**frequency**, where g ~ is

**the**effective coupling between

**the**resonator and

**, ω q and ω r are**

**qubit****the**angular

**resonance**frequencies

**of**

**the**

**and resonator, and σ z is ± 1 depending on**

**qubit****the**state

**of**

**the**

**. The composite signal probes all resonators at**

**qubit****the**same time, storing

**the**information on

**the**state

**of**all qubits in

**the**transmitted tones. Detection

**of**

**the**transmitted amplitude and phase

**of**each

**of**

**the**tones provides a simultaneous non-destructive measurement

**of**

**the**states

**of**all qubits. The probe signal is generated

**by**mixing a reference microwave tone in

**the**band

**of**

**the**resonators and a multi-tone DAC output using an IQ mixer, see Fig. fig:setup. By using

**the**I and Q quadratures, we address

**the**upper and lower sidebands

**of**

**the**mixing product individually to effectively double

**the**bandwidth

**of**

**the**system. The mixer output is combined with

**the**

**manipulation signal through a directional coupler. A strongly attenuated line transmits**

**qubit****the**combined signal to

**the**sample, which is attached to

**the**mixing chamber stage

**of**a dilution refrigerator. Two cryogenic circulators and a high-pass filter at 30 mK are used to prevent reflections and noise from traveling from

**the**cryogenic amplifier back to

**the**sample. A chain

**of**amplifiers provides 80 dB gain to boost

**the**transmitted probe signal to levels sufficient for

**the**detection stage, which employs an identical IQ mixer to convert

**the**signal back to baseband frequencies. The local oscillator inputs

**of**both mixers are fed from

**the**same reference microwave source, resulting in a homodyne detection with a fixed phase offset. An additional high-pass filter between

**the**local oscillator ports

**of**

**the**two mixers prevents leakage

**of**

**the**baseband signal. After digitizing both quadratures,

**the**amplitude and phase

**of**all components

**of**

**the**probe signal are extracted via FFT. The maximum number

**of**devices that can be probed with

**the**described technique is defined

**by**

**the**

**frequency**separation between resonators and

**the**bandwidth

**of**

**the**acquisition board. The frequencies

**of**

**the**resonators on our chip are spaced at intervals

**of**150 MHz and

**the**acquisition board has a bandwidth somewhat below 500 MHz, allowing for

**the**simultaneous detection

**of**up to six devices....(color online). Multiplexed spectroscopy of

**qubits**#2, 3 and 5. The

**qubit**manipulation microwave excites

**qubits**when its

**frequency**matches the transition between their ground and excited states. The state of all three

**qubits**is continuously and simultaneously monitored by the multi-tone probe signal. The horizontal axis reports the uniform bias flux applied to the chip....Experimental setup used for FDM readout. The

**qubit**manipulation signal is generated by a single microwave source for spectroscopy and three additional microwave sources, DAC channels and mixers for pulsed excitation of the

**qubits**....(color online). Multiplexed spectroscopy of

**qubits**#2, 3 and 5. The

**qubit**manipulation microwave excites

**qubits**when its

**frequency**matches the transition between their ground and excited states. The state of all three

**qubits**is continuously and simultaneously monitored by the multi-tone probe signal. The horizontal axis reports the uniform bias flux applied to the chip....superconducting flux

**qubit**,

**qubit**register, dispersive readout,

**frequency**division multiplexing, microwave resonators, Rabi

**oscillations**... An important desired ingredient of superconducting quantum circuits is a readout scheme whose complexity does not increase with the number of

**qubits**involved in the measurement. Here, we present a readout scheme employing a single microwave line, which enables simultaneous readout of multiple

**qubits**. Consequently, scaling up superconducting

**qubit**circuits is no longer limited by the readout apparatus. Parallel readout of 6 flux

**qubits**using a

**frequency**division multiplexing technique is demonstrated, as well as simultaneous manipulation and time resolved measurement of 3

**qubits**. We discuss how this technique can be scaled up to read out hundreds of

**qubits**on a chip.

Files:

Contributors: Rabenstein, K., Sverdlov, V. A., Averin, D. V.

Date: 2004-01-26

The profile of coherent quantum **oscillations** in an unbiased **qubit** dephased by the non-Gaussian noise with characteristic amplitude v 0 = 0.15 Δ and correlation time τ = 300 Δ -1 obtained by direct simulation of **qubit** dynamics with noise. Solid line is the exponential fit of the **oscillation** amplitude at large times. Dashed line is the initial 1 / t decay caused by effectively static distribution of v ....We have derived explicit non-perturbative expression for decoherence of quantum **oscillations** in a **qubit** by low-**frequency** noise. Decoherence strength is controlled by the noise spectral density at zero **frequency** while the noise correlation time $\tau$ determines the time $t$ of crossover from the $1/\sqrt{t}$ to the exponential suppression of coherence. We also performed Monte Carlo simulations of **qubit** dynamics with noise which agree with the analytical results and show that most of the conclusions are valid for both Gaussian and non-Gaussian noise....The rate γ of exponential **qubit** decoherence at long times t ≫ τ for ε = 0 and (a) Gaussian and (b) a model of the non-Gaussian noise with characteristic amplitude v 0 and correlation time τ . Solid lines give analytical results: Eq. ( e7) in (a) and Eq. ( e16) in (b). Symbols show γ extracted from Monte Carlo simulations of **qubit** dynamics. Note different scales for γ in parts (a) and (b). Inset in (b) shows schematic diagram of **qubit** basis states fluctuating under the influence of noise v t ....The rate γ of exponential **qubit** decoherence at long times t ≫ τ for ε = 0 and (a) Gaussian and (b) a model of the non-Gaussian noise with characteristic amplitude v 0 and correlation time τ . Solid lines give analytical results: Eq. ( e7) in (a) and Eq. ( e16) in (b). Symbols show γ extracted from Monte Carlo **simulations** of **qubit** dynamics. Note different scales for γ in parts (a) and (b). Inset in (b) shows schematic **diagram** of **qubit** basis states fluctuating under the influence of noise v t ....The profile of coherent quantum oscillations in an unbiased **qubit** dephased by the non-Gaussian noise with characteristic amplitude v 0 = 0.15 Δ and correlation time τ = 300 Δ -1 obtained by direct simulation of **qubit** dynamics with noise. Solid line is the exponential fit of the oscillation amplitude at large times. Dashed line is the initial 1 / t decay caused by effectively static distribution of v ....**Qubit** decoherence by low-**frequency** noise...We have derived explicit non-perturbative expression for decoherence of quantum oscillations in a **qubit** by low-**frequency** noise. Decoherence strength is controlled by the noise spectral density at zero **frequency** while the noise correlation time $\tau$ determines the time $t$ of crossover from the $1/\sqrt{t}$ to the exponential suppression of coherence. We also performed Monte Carlo simulations of **qubit** dynamics with noise which agree with the analytical results and show that most of the conclusions are valid for both Gaussian and non-Gaussian noise. ... We have derived explicit non-perturbative expression for decoherence of quantum **oscillations** in a **qubit** by low-**frequency** noise. Decoherence strength is controlled by the noise spectral density at zero **frequency** while the noise correlation time $\tau$ determines the time $t$ of crossover from the $1/\sqrt{t}$ to the exponential suppression of coherence. We also performed Monte Carlo simulations of **qubit** dynamics with noise which agree with the analytical results and show that most of the conclusions are valid for both Gaussian and non-Gaussian noise.

Files:

Contributors: Leyton, V., Thorwart, M., Peano, V.

Date: 2011-09-26

Before turning to **the** quantum detection scheme, we discuss **the** dynamical properties of **the** isolated detector, which is **the** quantum Duffing oscillator. A key property is its nonlinearity which generates multiphoton transitions at frequencies ω e x close to **the** fundamental **frequency** Ω . In order to see this, one can consider first **the** undriven** nonlinear** oscillator with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , **the** degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in **the** detector. As a consequence, **the** amplitude A of **the** nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small oscillation state is predominantly populated. The formation of peaks and dips goes along with jumps in **the** phase of **the** oscillation, leading to oscillations in or out of phase with **the** driving. A typical example of **the** nonlinear response of **the** quantum Duffing oscillator in **the** deep quantum regime containing few-photon (anti-)resonances is shown in Fig. fig1(a) (decoupled from **the** **qubit**), together with **the** corresponding quasienergy spectrum [Fig. fig1(b)]. We show **the** multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined **by ****the** Rabi **frequency**, which is given **by ****the** minimal splitting at **the** corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to **the** driving strength f , one can get **the** minimal energy splitting at **the** avoided quasienergy level crossing 0 N as...(Color online) (a) Asymptotic population difference P ∞ of the **qubit** states, and (b) the corresponding detector response A as a function of the external **frequency** ω e x for the same parameters as in Fig. fig2. fig4...(Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external **frequency** ω e x . The parameters are the same as in Fig. fig2. fig5...(Color online) (a) Relaxation rate Γ of **the** nonlinear quantum detector, (b) **the** measurement time T m e a s , and (c) **the** measurement efficiency Γ m e a s / Γ as a function of **the** external **frequency** ω e x . The parameters are **the** same as in Fig. fig2. fig5...(Color online) Nonlinear response A of **the** detector as a function of **the** external driving **frequency** ω e x in **the** presence of a finite coupling g = 0.0012 Ω to **the** **qubit** (black solid line). The blue dashed line indicates **the** response of **the** isolated detector. The parameters are **the** same as in Fig. fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2...For a fixed value of g , the shift between the two cases of the opposite **qubit** states is given by the **frequency** gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the **qubit** is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line)....We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed....(Color online) (a) Amplitude A of **the** nonlinear response of **the** decoupled quantum Duffing detector ( g = 0 ) as a function of **the** external driving **frequency** ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote **the** corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1...**Qubit** state detection using the quantum Duffing **oscillator**...Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired **oscillator** **frequency** Ω , where the coupling term is considered as a perturbation to the SQUID ( g **oscillator** to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig. fig0....(Color online) (a) Nonlinear response A of the detector coupled to the **qubit** prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig. fig2. The quadratic **qubit**-detector coupling induces a global **frequency** shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the **qubit** for the same parameters as in a). fig3...For a fixed value of g , **the** shift between **the** two cases of **the** opposite **qubit** states is given **by ****the** **frequency** gap δ ω e x ≃ 2 g . Figure fig3 (a) shows **the** nonlinear response of **the** detector for **the** two cases when **the** **qubit** is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line)....(Color online) (a) Nonlinear response A of **the** detector coupled to **the** **qubit** prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for **the** same parameters as in Fig. fig2. The **quadratic** **qubit**-detector coupling induces a **global** **frequency** shift of **the** response by δ ω e x = 2 g . (b) Discrimination power D ω e x of **the** detector coupled to **the** **qubit** for **the** same parameters as in a). fig3...For a rough evaluation of **the** order of magnitude of **the** involved time scales, we may neglect **the** nonlinearity of **the** detector ( α = 0 ) for **the** moment and estimate **the** effective relaxation rate for **the** **qubit** coupled to an Ohmically damped harmonic oscillator. This model can be mapped to a **qubit** coupled to a structured harmonic environment with an effective (dimensionless) coupling constant κ e f f = 8 γ g 2 / Ω 2 . For **the** realistic parameters used in Fig. fig1 and g = 0.0012 Ω , we find that κ e f f ≃ 10 -10 , giving rise to an estimated relaxation rate Γ h a r m ≃ π / 2 sin 2 θ κ e f f ϵ ≃ 10 -13 Ω (evaluated at low temperature). Hence, this illustrates that we can easily achieve **the** situation where Γ h a r m ≪ γ required for this detection scheme. Then, for a waiting time (after which we start **the** measurement) much longer than **the** relaxation time γ -1 of **the** nonlinear oscillator, but still smaller than Γ...(Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving **frequency** ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1...(Color online) (a) Nonlinear response A of **the** detector coupled to **the** **qubit** prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for **the** same parameters as in Fig. fig2. The **quadratic** **qubit**-detector coupling induces a **global** **frequency** shift of **the** response **by **δ ω e x = 2 g . (b) Discrimination power D ω e x of **the** detector coupled to **the** **qubit** for **the** same parameters as in a). fig3...Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing **oscillator**. A key property is its nonlinearity which generates multiphoton transitions at **frequencies** ω e x close to the fundamental **frequency** Ω . In order to see this, one can consider first the undriven nonlinear **oscillator** with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small **oscillation** state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the **oscillation**, leading to **oscillations** in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing **oscillator** in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig. fig1(a) (decoupled from the **qubit**), together with the corresponding quasienergy spectrum [Fig. fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi **frequency**, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as...(Color online) Nonlinear response A of the detector as a function of the external driving **frequency** ω e x in the presence of a finite coupling g = 0.0012 Ω to the **qubit** (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig. fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2...(Color online) (a) Asymptotic population difference P ∞ of **the** **qubit** states, and (b) **the** corresponding detector response A as a function of **the** external **frequency** ω e x for **the** same parameters as in Fig. fig2. fig4 ... We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed.

Files:

Contributors: Hausinger, Johannes, Grifoni, Milena

Date: 2010-09-08

Figure Fig::SpectrumVSg shows the quasienergy spectrum against the coupling strength g . For simplicity, we study the unbiased case ε = 0 , which implies m = L = 0 and hence gaps with Ω 0 , 0 n , K = | Δ 0 L K 0 α e - α 2 | ≡ Ω K . Thus, for g = 0 and Δ ≠ 0 , the twofold degeneracy of the unperturbed case is lifted by a gap of width Δ 0 . For g ≠ 0 , the gap size is further determined by the Laguerre polynomial, so that additional degeneracies can occur at the zeros of L K 0 α . When choosing the driving amplitude A such that Δ 0 = 0 the twofold degeneracy is kept for arbitrary g and K . Because the dressing by the Bessel function does not depend on g or the **oscillator** level, we reach the remarkable conclusion that the coherent destruction of tunneling (CDT), predicted for a driven **qubit** , might occur also for a **qubit**-**oscillator** system in the ultrastrong coupling limit. In Fig. Fig::DressedOsc, the dressed oscillation **frequencies** are plotted against the dimensionless coupling g / Ω . Next to an exponential decay, they exhibit zeros that depend through the Laguerre polynomial characteristically on the **oscillator** quantum number K . Hence, because the **qubit**’s dynamics involves several **oscillator** levels, we predict that suppression of tunneling cannot be reached by just tuning the coupling g . The dynamics. To prove the statements above, we calculate the survival probability of the **qubit** P ↓ ↓ t : = ↓ | ρ ̂ r e d t | ↓ , where ρ ̂ r e d is obtained by tracing out the **oscillator** degrees of freedom from the density operator of the **qubit**-**oscillator** system:...In Fig. Fig::Dynam1(c) we are with g / Ω = 1.0 already deep in the ultrastrong coupling regime. The **frequency** Ω 1 is now different from zero, and additionally Ω 3 appears. The lowest peak belongs to the **frequencies** Ω 0 , Ω 2 , and Ω 4 , which are equal for g / Ω = 1.0 , see Fig. Fig::DressedOsc. A complete population inversion again takes place. Our results are confirmed by numerical calculations. For g = 0.5 , 1.0 , the latter yield additionally fast **oscillations** with Ω and ω ex . Furthermore, Ω 1 is shifted in Fig. Fig::Dynam1(c) slightly to the left, so that concerning the survival probability the analytical and numerical curves get out of phase for longer times. To include also the **oscillations** induced by the driving and the coupling to the quantized modes, connections between the degenerate subspaces need to be included in the calculation of the eigenstates of the full Hamiltonian ....(Color online) Size of the avoided crossing Ω K against the dimensionless coupling strength g / Ω for an unbiased **qubit** ( ε = 0 ). Further, Δ / Ω = 0.4 , ω ex / Ω = 5.3 and A / Ω = 8.0 . Ω K vanishes at the zeros of the Laguerre polynomial L K 0 α . The dashed lines (a), (b), (c) represent g / Ω = 0.1 , 0.5 , 1.0 , respectively, as considered in Fig. Fig::Dynam1. Fig::DressedOsc...(Color online) Coherent destruction of tunneling in a driven **qubit**-**oscillator** system. The same parameters as in Fig. Fig::Dynam1 are used except that A / Ω = 12.7 , which leads to Δ 0 = 0 . Three coupling strengths are examined: g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). The analytical calculations (black, dashed lines) predict complete localization for all three cases. Also the numerics (red curves) shows strong localization for short timescales with fast oscillations overlaid. For long times this localization vanishes (see inset in (a)). Fig::DynamCDT...**Qubit**-**oscillator** system under ultrastrong coupling and extreme driving...While tuning the coupling g to a zero of a Laguerre polynomial corresponding to a dominant **oscillator** mode yields a reduction of tunneling, tuning the driving amplitude A to a zero of a Bessel function can yield almost complete localization at short times. As already noticed in Fig. Fig::SpectrumVSg, this phenomenon is independent of the coupling strength g . We choose in Fig. Fig::DynamCDT the driving amplitude A , so that Δ 0 = 0 . This is the same condition as found for CDT in a driven **qubit** . Analogously, our analytical solution now predicts localization for arbitrary coupling strength g . All dressed **oscillation** **frequencies** Ω K vanish. However, third-order corrections in Δ will give small contributions to Δ 0 . Hence, a numerical exact solution yields **oscillations** of P ↓ ↓ t with a long period. On a short timescale and for ω ex ≫ Δ also the numerical solution appears to be strongly localized, while for long times, the inset in Fig. Fig::DynamCDT (a) shows complete population inversion for the numerics. In conclusions, we developed a powerful formalism to investigate analytically a **qubit**-**oscillator** system in the ultrastrong coupling and extreme driving regime, a situation which is in close experimental reach and offers excellent control possibilities. Our approach relies on perturbation theory with respect to a single parameter only, the **qubit** tunneling matrix element Δ , and thus goes beyond the driven Jaynes-Cummings model, with no rotating-wave approximation being applied. We acknowledge financial support under DFG Program SFB631. We thank Sigmund Kohler for helpful remarks....(Color online) Dynamics of the **qubit** for ε = 0 , Δ / Ω = 0.4 , ω ex / Ω = 5.3 , A / Ω = 8.0 , and temperature ℏ Ω k B T -1 = 10 . The graphs show the Fourier transform F ν of the survival probability P ↓ ↓ t (see the insets). We study the different coupling strengths indicated in Fig. Fig::DressedOsc, g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). Analytical results are shown by black curves, numerics by dashed orange curves. Fig::Dynam1...Additional crossings occur independent of ε if driving and **oscillator** frequency are commensurable, Ω / ω ex = j / N with integers j , N > 0 , resulting in infinite many degenerate states. We avoid such a situation by choosing incommensurable **frequencies** or high values for j and N , so that only high-photon processes are affected.. Note that for L ≠ 0 there are always L nondegenerate levels. For L > 0 those are the first L spin-down states (positive slope), while for L < 0 the first L spin-up states (negative slope). At finite Δ avoided crossings occur in the energy spectrum at the sites of the resonances (red triangles and black dots in Fig. Fig::QuasiEnEpsAnaDfinite). To explain the origin of these avoided crossings we express H ̂ in the basis ( CoupledEigenstates) yielding the off-diagonal elements...Figure Fig::SpectrumVSg shows the quasienergy spectrum against the coupling strength g . For simplicity, we study the unbiased case ε = 0 , which implies m = L = 0 and hence gaps with Ω 0 , 0 n , K = | Δ 0 L K 0 α e - α 2 | ≡ Ω K . Thus, for g = 0 and Δ ≠ 0 , the twofold degeneracy of the unperturbed case is lifted by a gap of width Δ 0 . For g ≠ 0 , the gap size is further determined by the Laguerre polynomial, so that additional degeneracies can occur at the zeros of L K 0 α . When choosing the driving amplitude A such that Δ 0 = 0 the twofold degeneracy is kept for arbitrary g and K . Because the dressing by the Bessel function does not depend on g or the **oscillator** level, we reach the remarkable conclusion that the coherent destruction of tunneling (CDT), predicted for a driven **qubit** , might occur also for a **qubit**-**oscillator** system in the ultrastrong coupling limit. In Fig. Fig::DressedOsc, the dressed **oscillation** **frequencies** are plotted against the dimensionless coupling g / Ω . Next to an exponential decay, they exhibit zeros that depend through the Laguerre polynomial characteristically on the **oscillator** quantum number K . Hence, because the ** qubit’s** dynamics involves several

**oscillator**levels, we predict that suppression of tunneling cannot be reached by just tuning the coupling g . The dynamics. To prove the statements above, we calculate the survival probability of the

**qubit**P ↓ ↓ t : = ↓ | ρ ̂ r e d t | ↓ , where ρ ̂ r e d is obtained by tracing out the

**oscillator**degrees of freedom from the density operator of the

**qubit**-

**oscillator**system:...We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling....(Color online) Quasienergy spectrum of the

**qubit**-

**oscillator**system against the static bias ε for weak coupling g / ω ex = 0.05 . Further parameters are Δ / ω ex = 0.2 , Ω / ω ex = 2 , A / ω ex = 2.0 . The first six

**oscillator**states are included. Numerical calculations are shown by red (light gray) triangles, analytical results in the region of avoided crossings by black dots. A good agreement between analytics and numerics is found. Blue (dark gray) squares represent the case Δ = 0 . Fig::QuasiEnEpsAnaDfinite...While tuning the coupling g to a zero of a Laguerre polynomial corresponding to a dominant

**oscillator**mode yields a reduction of tunneling, tuning the driving amplitude A to a zero of a Bessel function can yield almost complete localization at short times. As already noticed in Fig. Fig::SpectrumVSg, this phenomenon is independent of the coupling strength g . We choose in Fig. Fig::DynamCDT the driving amplitude A , so that Δ 0 = 0 . This is the same condition as found for CDT in a driven

**qubit**. Analogously, our analytical solution now predicts localization for arbitrary coupling strength g . All dressed oscillation

**frequencies**Ω K vanish. However, third-order corrections in Δ will give small contributions to Δ 0 . Hence, a numerical exact solution yields oscillations of P ↓ ↓ t with a long period. On a short timescale and for ω ex ≫ Δ also the numerical solution appears to be strongly localized, while for long times, the inset in Fig. Fig::DynamCDT (a) shows complete population inversion for the numerics. In conclusions, we developed a powerful formalism to investigate analytically a

**qubit**-

**oscillator**system in the ultrastrong coupling and extreme driving regime, a situation which is in close experimental reach and offers excellent control possibilities. Our approach relies on perturbation theory with respect to a single parameter only, the

**qubit**tunneling matrix element Δ , and thus goes beyond the driven Jaynes-Cummings model, with no rotating-wave approximation being applied. We acknowledge financial support under DFG Program SFB631. We thank Sigmund Kohler for helpful remarks....(Color online) Coherent destruction of tunneling in a driven

**qubit**-

**oscillator**system. The same parameters as in Fig. Fig::Dynam1 are used except that A / Ω = 12.7 , which leads to Δ 0 = 0 . Three coupling strengths are examined: g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). The analytical calculations (black, dashed lines) predict complete localization for all three cases. Also the numerics (red curves) shows strong localization for short timescales with fast

**oscillations**overlaid. For long times this localization vanishes (see inset in (a)). Fig::DynamCDT...The dressed tunneling matrix element in Eq. ( HTLSHOeff) determines to first order the width of the avoided crossings in Fig. Fig::QuasiEnEpsAnaDfinite. Dominant crossings are found for ε = m ω ex , where the static bias is an integer multiple of the driving frequency, and thus L = 0 . That means that both states belong to the same

**oscillator**quantum number K , and the dressing contains a Laguerre polynomial of the kind L K 0 α . For the 2 × 2 block in Eq. ( HTLSHOeff) the eigenvalues to the eigenstates | Φ m , L ∓ , n , K are found easily:...Additional crossings occur independent of ε if driving and

**oscillator**

**frequency**are commensurable, Ω / ω ex = j / N with integers j , N > 0 , resulting in infinite many degenerate states. We avoid such a situation by choosing incommensurable

**frequencies**or high values for j and N , so that only high-photon processes are affected.. Note that for L ≠ 0 there are always L nondegenerate levels. For L > 0 those are the first L spin-down states (positive slope), while for L < 0 the first L spin-up states (negative slope). At finite Δ avoided crossings occur in the energy spectrum at the sites of the resonances (red triangles and black dots in Fig. Fig::QuasiEnEpsAnaDfinite). To explain the origin of these avoided crossings we express H ̂ in the basis ( CoupledEigenstates) yielding the off-diagonal elements ... We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling.

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